Abstract: The main aim of the study is to provide the generalized stochastic perturbation technique based on the classical Taylor expansion with a single random variable. The main problem discussed below is an application of this expansion to the solution of various partial differential equations with random coefficients by the fundamental numerical methods, i.e., Boundary Element Method, Finite Element Method as well as the Finite Difference Method. Since, nth order expansion is employed for this purpose, the probabilistic moments of the solution can be determined with a priori given accuracy. Contrary to the second order techniques used before, a perturbation parameter is also included in the relevant approximations, so the overall solution convergence can be speeded up by some modification of its value. Application of computational methodologies presented in transient problems (dynamics or heat transfer) are also commented in the study together with stochastic processes modelling by the double Taylor expansion.